In Lesson 3.1.1, you used the meaning of an exponent to rewrite expressions such as y⁴ · y ² and (x²y)³.  Today you will use the patterns you discovered to learn how to interpret expressions with exponents that are negative or zero.

• 3-12. Review what you learned about exponents in Lesson 3.1.1 to rewrite each expression below as simply as possible.  If you see a pattern or know of a shortcut, be sure to share it with your teammates.
  1. x⁷ · x ⁴
  2. (x³)³
  3. 
  4. (x²y ²)⁴
  5. 
  6. 
• 3-13. With your study team, summarize the patterns you found in problem 3-12.  For each one, simplify the given expression and write an expression that represents its generalization.  Then, in your own words, explain why the pattern works.

  	Expression	Generalization	Why is this true?
  a.	x25 · x40 = ?	xm · xn = ?
  b.
  c.	(x5)12 = ?	(xm)n = ?

• 3-14. Describe everything you know about . What is its value?  How can you rewrite it using a single exponent?  What new conclusions can you draw?  Be prepared to explain your findings to the class. 
• 3-15. Problem 3-14 helped you recognize that  x⁰ = 1. Now you will similarly use division to explore the meaning of  x⁻¹, x⁻², etc. Simplify each of the expressions below twice:
  • Once by expanding the terms and simplifying.
  • Again by using your new pattern for division with exponents.
  
  Be ready to discuss the meaning of negative exponents with the class. 
  1. 
  2. 
  3. 
• 3-16. Use your exponent patterns to rewrite each of the expressions below.  For example, if the original expression has a negative exponent, then rewrite the expression so that it has no negative exponents, and vice versa.  Also, if the expression contains multiplication or division, then use your exponent rules to simplify the expression.
  1. k⁻⁵
  2. m⁰
  3. x⁻² · x⁵
  4. 
  5. 
  6. (x⁻²)³
  7. (a²b)⁻¹
  8. 
• 3-17. EXPONENT CONCENTRATION

  Split your team into two pairs and decide which is Team A and which is Team B.  Your teacher will distribute a set of cards for a game described below.

  • Arrange the cards face down in a rectangular grid.
  • Team A selects and turns over two cards. 
  • If Team A thinks the values on the cards are equivalent, they must justify this claim to Team B.  If everyone in Team B agrees, Team A takes the pair.  If the values are not equivalent, Team A returns both cards to their original position (face down).  This is the end of the turn for Team A.
  • Team B repeats the process.
  • Teams alternate until no cards remain face down.  The team with the most matches wins.
• 3-18.  In your Learning Log, describe the meaning of zero and negative exponents.  That is, explain how to interpret  x⁰ and  x⁻¹. Title this entry “Zero and Negative Exponents” and include today’s date.
  
  
   

Laws of Exponents

In the expression x³,  x is the base and 3 is the exponent.

x³ = x · x · x

The patterns that you have been using during this section of the book are called the laws of exponents.  Here are the basic rules with examples: